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# Evaluating Researcher's Claim using a Two-Tailed T-Test

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A researcher is interested in comparing the response times of two different cab companies. Companies A and B were each called at 50 randomly selected times. The calls to company A were independently made from the calls to company B. Company A had a mean response time of 8.5 minutes with a standard deviation of 1.8 minutes. Company B had a mean response time of 5.5 minutes with a standard deviation of 1.6 minutes. Use a 0.05 significance level to test the claim that the mean response time for company A differs from the mean response time for company B by doing the following:

1. State the null and alternative hypotheses symbolically and identify which represents the claim.
2. Determine whether the hypothesis test is a one-tailed test or a two-tailed test and whether or not to use a z-test or a t-test.
3. Identify the critical value(s) and identify the rejection region(s).
4. Use the appropriate test to find the appropriate test statistic. You can use technology for this question but please attach your results which must show your work.
5. Based on your test statistic and your rejection region, what is your decision?
6. Interpret your decision in context of the original claim.
7. Construct a 95% confidence interval estimate of the difference between the mean responses times for Company A and Company B. Interpret your interval in context of the original claim.

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#### Solution Preview

Hi there,

1. Null hypothesis: the mean response time for company A is same as the mean response time for company B. in other words, µA=µB
Alternative hypothesis: the mean response time for company A is different from the mean response time for company B. in other words, µA?µB.
The claim is same as the alternative hypothesis.

2. Since the sample size is bigger than 30 and ...

#### Solution Summary

The solution provides detailed explanation how to perform two tailed t test assuming equal variance.

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## NCSS assignment: hypothesis testing and regression.

Use NCSS to conduct all statistical tests and calculations. For each question, you are to edit, copy, paste and highlight the appropriate output from NCSS and produce a typed document answering the questions. Be succinct. Control the Type I error rate at the 0.05 level for all statistical test procedures and confidence intervals. At a minimum, for all hypothesis tests, be sure to:
1. State the null and alternate hypotheses
2. Choose a statistical test and calculate the test statistic
3. Calculate the p-value of the test statistic
4. Make your statistical decision
5. State your conclusion

Problem 1. Height and weight are often used in epidemiological studies as possible predictors of disease outcomes. If patients are assessed in the clinic, then heights and weights are usually measured directly. However, if people are interviewed by phone or mail, then a person's self-reported height and weight are often used instead. Suppose we conduct a study on 10 people to test the comparability of these two methods as to "location" of the population distributions of weight. Conduct a nonparametric statistical test to answer the study question. The weight data follows.

ID # Self-reported Weight (lbs.) Measured Weight (lbs.)
1 120 125
2 120 118
3 135 139
4 118 120
5 120 125
6 190 198
7 124 128
8 175 176
9 133 131
10 125 125

Problem 2. Total heart weight (THW) was measured at autopsy on a group of 11 males with left-heart disease and 10 normal males. Test for a significant difference in THW between the two groups using a nonparametric statistical procedure. The autopsy data follows.

Left-Heart Disease Males Normal Males
ID # THW (g) ID # THW (g)
1 450 12 245
2 760 13 350
3 325 14 340
4 495 15 300
5 285 16 310
6 450 17 270
7 460 18 300
8 375 19 360
9 310 20 405
10 615 21 290
11 425
Problem 3. A study was conducted focusing on the protein concentration of duodenal secretions from patients with cystic fibrosis. The following table relates protein concentration (mg/ml) to pancreatic function as measured by trypsin secretion.

Trypsin Secretions [u/(kg/hr)]
£ 50 51-1000 ³ 1001
Subject Number Protein Concentration Subject Number Protein Concentration Subject Number Protein Concentration
1 1.7 10 1.4 20 2.9
2 2.0 11 2.4 21 3.8
3 2.0 12 2.4 22 4.4
4 2.2 13 3.3 23 4.7
5 4.0 14 4.4 24 5.0
6 4.0 15 4.7 25 5.6
7 5.0 16 6.7 26 7.4
8 6.7 17 7.6 27 9.4
9 7.8 18 9.5 28 10.3
19 11.7

a) What parametric and nonparametric statistical procedures can be used to compare the protein concentration for the three groups?
b) Perform both tests mentioned above.
c) Compare all pairs of group means/medians using both parametric and nonparametric methods. Use a Bonferroni correction to insure a family-wise error rate of no more than 0.05.

Problem 4. The following table gives contraceptive usage among a simple random sample of 100 women age 15 - 45 discharged from a large hospital with a diagnosis of idiopathic thromboembolism. A researcher wanted to know if there is a difference in contraceptive usage among women suffering from thromboembolic disease. Conduct a statistical test to answer the researcher's question.

Contraceptive Used Observed Count
Oral Contraceptive 30
IUD 25
Diaphragm 20
None 25
Total 100

Problem 5. The following tabulation represents the outcome of a drug trial of 60 patients assigned to a new treatment as compared to 40 patients assigned to the standard treatment for a particular disease. The researcher wanted to know if the new treatment was different than the standard treatment. Conduct a statistical test to answer the researcher's question.

Treatment Group Improved Not Improved Total
New 40 20 60
Standard 25 15 40
Total 65 35 100

Problem 6. Sleep-disordered breathing is very common among adult males. Snoring is a significant contributor to these disorders, in particular with regards to sleep apnea. To estimate the prevalence of this disorder 1670 male employees, 30-60 years of age, who worked for three large state agencies in Georgia were surveyed as to their snoring status. Conduct a statistical test to assess if snoring status is related to age. The results of the survey are given in the following table.

Age Snore Total
Yes No
30-39 188 348 536
40-49 313 383 696
50-60 232 206 438
Total 733 937 1670

Problem 7. A company operates a production line for making prescription pill bottles. The relation between the speed of the line (X) and the amount of scrap (Y) for the day was studied for 15 days. The basic data for the line is given in the table below.

Day
Y (lbs.) X (ft./min.)
1 218 100
2 248 125
3 360 220
4 351 205
5 470 300
6 394 255
7 332 225
8 321 175
9 410 270
10 260 170
11 241 155
12 331 190
13 275 140
14 425 290
15 367 265

a) For these data obtain estimates for the regression parameters, a and b, for the simple linear regression of Y on X and interpret the meaning of each estimator.

b) What is the fitted regression line?

c) Find the standard error of each estimator in part a.

d) Set a 95% confidence interval on a and b and interpret the meaning of these intervals.

e) Write out the ANOVA for the regression giving source, degrees-of-freedom, sum-of-squares, mean squares.

f) What is the estimated variance of the regression (s2Y|X)?

g) Perform an F-test for the "significance" of regression. Be sure to clearly state the hypotheses tested and your conclusions.

h) Compute R2 and interpret its meaning.

i) Compute Pearson's product-moment (simple) correlation coefficient (r). What is its associated p-value. Interpret its meaning and assess its significance.

j) Compute Spearman's rank-order correlation coefficient (rs). What is its associated p-value. Interpret its meaning and assess its significance.

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